Fractional Calculus and Applied Analysis最新文献

In this paper, the existence of weak solutions to some Dirichlet problems with fractional competing operators and distributional Riesz fractional gradient is investigated. Due to the nature of driving operators, the most known techniques, basically based on ellipticity and monotonicity, are no longer applicable. Generalized solutions (in a suitable sense) are obtained via an approximation procedure and a corollary of the Brouwer fixed point theorem.

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of -convolution.

A systematic investigation of the significance and applicability of two different approaches of generalized separation of variable (GSV) methods for time-fractional nonlinear PDEs in ((2+1)) and ((m+1))-dimensions is presented. Also, this work explicitly shows that while constructing the exact solutions of time-fractional nonlinear PDEs in ((2+1)) and ((m+1))-dimensions without and with delay terms, how to overcome unusual (non-standard) properties of singular kernel fractional derivatives such as chain rule, semigroup property, and the Leibniz rule. Moreover, the importance and effectiveness of the two GSV methods have been discussed through the initial and boundary value problems of the time-fractional nonlinear generalized convection-diffusion equation in ((2+1))-dimensions. Additionally, the discussed methods extended to find the exact solutions of time-fractional nonlinear PDEs in ((2+1)) and ((m+1))-dimensions involving multiple linear time-delay terms along with appropriate examples. Also, this work investigates the comparative study of the obtained results and solutions of the underlying equations using the two GSV methods, along with the 2D and 3D graphical representations.

The paper deals with a source identification problem of the anomalous diffusion equations from nonlocal final data observations where the nonlinearity probably takes values in Hilbert scales. The existence and uniqueness results are proved by establishing some estimates for resolvent operators and using the embedding theorems. We also study regularity results for this equation in terms of the Hölder continuity of mild solutions. Finally, the multi-term fractional diffusion equations with polynomial nonlinearities and the ultra-slow diffusions are considered as illustrative applications.

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, (mathcal {H}_{hbar ,V}:=-hbar ^{-2}mathcal {L}_{hbar }+V) on the lattice (hbar mathbb {Z}^{n},) where V is a positive multiplication operator and (mathcal {L}_{hbar }) is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit (hbar rightarrow 0).

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered.

This paper study a class of time-space fractional reaction-diffusion equations with nonlocal initial conditions and construct an abstract theory in fractional power spaces to discuss the results related S-asymptotically (omega )-periodic mild solutions. When the coefficients are sufficiently small, under the condition that the nonlinear term can grow any number of orders, we discuss the existence and uniqueness of S-asymptotically (omega )-periodic solutions based on the theory of operator semigroups and fixed point theorem. In addition, we considered the Mittag-Leffler-Ulam-Hyers stability results using the singular type Gronwall inequality and appropriate fractional calculus. The results in the present paper extended the work of (Andrade et al. in Proc Edinb Math Soc 59:65–76, 2016) to the case of time-space fractional nonlocal reaction-diffusion equations.

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique.

The present paper is concerned with a fractional parabolic reaction-diffusion system posed in a regular bounded open subset of ({mathbb {R}}^N), where the gradients of the unknowns act as source terms (see (S) below). First, we establish some nonexistence and blow-up in finite time results. Second, we prove some new weighted regularity results. Such results are interesting in themselves and play a crucial role to study local existence of nonnegative solutions to our system under suitable assumptions on the data. This work also highlights a substantial difference between the nonlocal case and the local case already studied by the fourth author and his coworkers.